Logica Universalis

, Volume 2, Issue 1, pp 167–187

Logical Extensions of Aristotle’s Square

  • Dominique Luzeaux
  • Jean Sallantin
  • Christopher Dartnell

DOI: 10.1007/s11787-007-0022-y

Cite this article as:
Luzeaux, D., Sallantin, J. & Dartnell, C. Log. univers. (2008) 2: 167. doi:10.1007/s11787-007-0022-y


We start from the geometrical-logical extension of Aristotle’s square in [6,15] and [14], and study them from both syntactic and semantic points of view. Recall that Aristotle’s square under its modal form has the following four vertices: A is □α, E is \(\square\neg\alpha\), I is \(\neg\square\neg\alpha\) and O is \(\square\neg\alpha\), where α is a logical formula and □ is a modality which can be defined axiomatically within a particular logic known as S5 (classical or intuitionistic, depending on whether \(\neg\) is involutive or not) modal logic.

[3] has proposed extensions which can be interpreted respectively within paraconsistent and paracomplete logical frameworks. [15] has shown that these extensions are subfigures of a tetraicosahedron whose vertices are actually obtained by closure of \(\{\alpha,\square\alpha\}\) by the logical operations \(\{\neg,\wedge,\vee\}\), under the assumption of classical S5 modal logic. We pursue these researches on the geometrical-logical extensions of Aristotle’s square: first we list all modal squares of opposition. We show that if the vertices of that geometrical figure are logical formulae and if the sub-alternation edges are interpreted as logical implication relations, then the underlying logic is none other than classical logic. Then we consider a higher-order extension introduced by [14], and we show that the same tetraicosahedron plays a key role when additional modal operators are introduced. Finally we discuss the relation between the logic underlying these extensions and the resulting geometrical-logical figures.

Mathematics Subject Classification (2000).

Primary 03B45 Secondary 06F99 


Aristotle’s square modal logic 

Copyright information

© Birkhaeuser 2008

Authors and Affiliations

  • Dominique Luzeaux
    • 1
  • Jean Sallantin
    • 2
  • Christopher Dartnell
    • 2
  1. 1.DSA/UMSIOBagneux CedexFrance
  2. 2.LIRMM, UMR 5506Montpellier Cedex 5France